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Question

If fx=xsinπx+1+sinπx+11+x, then


A

fx is continuous on .

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B

fx is continuous on , but not differentiable on

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C

fx is discontinuous at every integer.

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D

None of these.

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Solution

The correct option is C

fx is discontinuous at every integer.


Explanation for the correct option:

Step 1: Simplify the given expression

The given function is fx=xsinπx+1+sinπx+11+x.

Let x+1=n, where n.

Thus, sinπx+1=sinπn

sinπx+1=0.

So, fx=xsinπx+11+x.

Step 2: Check for continuity

Let us assume that n is an integer.

Thus, limxn-fx=limxn-xsinπx+11+x

limxn-fx=n-1sinπnnlimxn-fx=n-1nsinπn

Thus, limxn+fx=limxn+xsinπx+11+x

limxn+fx=nsinπn+11+nlimxn+fx=nn+1sinπn+1

Thus, limxn-fxlimxn+fx.

Therefore, fx is discontinuous at every integer.

Hence, option(C) i.e. fx is discontinuous at every integer is the correct option.


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